c 2004 Society for Industrial and Applied Mathematics
SIAM J. APPL. MATH.
Vol. 65, No. 1, pp. 336–360
SINGULARITY FORMATION IN CHEMOTAXIS—
A CONJECTURE OF NAGAI∗
HOWARD A. LEVINE† AND JOANNA RENCLAWOWICZ† ‡
Abstract. Consider the initial-boundary value problem for the system (S) ut = uxx − (uvx )x ,
vt = u − av on an interval [0, 1] for t > 0, where a > 0 with ux (0, t) = ux (1, t) = 0. Suppose
µ, v0 are positive constants. The corresponding spatially homogeneous global solution U (t) = µ,
V (t) = µ/a + (v0 − µ/a) exp(−at) is stable in the sense that if (µ , v0 ) are positive constants, the
corresponding spatially homogeneous solution will be uniformly close to (U (·), V (·)).
We consider, in sequence space, an approximate system (S ) which is related to (S) in the
following sense: The chemotactic term (uvx )x is replaced by the inverse Fourier transform of the
finite part of the convolution integral for the Fourier transform of (uvx )x . (Here the finite part of
x
the convolution on the line at a point x of two functions, f, g, is defined as
(f (y)g(y − x) dy.) We
0
prove the following:
(1) If µ > a, then in every neighborhood of (µ, v0 ) there are (spatially nonconstant) initial
data for which the solution of problem (S ) blows up in finite time in the sense that the
solution must leave L2 (0, 1) × H 1 (0, 1) in finite time T . Moreover, the solution components
u(·, t), v(·, t) each leave L2 (0, 1).
(2) If µ > a, then in every neighborhood of (µ, v0 ) there are (spatially nonconstant) initial
data for which the solution of problem (S) on (0, 1) × (0, Tmax ) must blow up in finite time
in the sense that the coefficients of the cosine series for (u, v) become unbounded in the
sequence product space 1 × 11 .
A consequence of (2) states that in every neighborhood of (µ, v0 ), there are solutions of (S)
which, if they are sufficiently regular, will blow up in finite time. (Nagai and Nakaki [Nonlinear
Anal., 58 (2004), pp. 657–681] showed that for the original system such solutions are unstable in the
sense that if µ > a, then in every neighborhood of (µ, µ/a), there are spatially nonconstant solutions
which blow up in finite or infinite time. They conjectured that the blow-up time must be finite.)
Using a recent regularity result of Nagai and Nakaki, we prove this conjecture.
Key words. chemotaxis, finite time singularity formation, Keller–Segel model
AMS subject classifications. 35K55, 92C17
DOI. 10.1137/S0036139903431725
1. Introduction. The classical equations of chemotaxis were introduced in [15,
16, 22]. A variant of them, which was later discussed in [14], takes the form
(1.1)
ut = D1 (∆u − ∇ · (uχ(v)∇v)),
vt = D2 ∆v + λu − av
when the diffusivities Dj are constant. The constants in (1.1) are presumed to be
positive. Generally speaking, u(·, t) represents the cell concentration (or local population) of some species, while v(·, t) corresponds to the concentration of a chemotactic
agent such as cyclic adenosine monophosphate (cAMP). The function χ(v) is called
the chemotactic sensitivity.
∗ Received by the editors July 14, 2003; accepted for publication (in revised form) January 20, 2004;
published electronically October 28, 2004. The first author was supported in part by Mathematical
Sciences Biology Institute of Ohio State University. This material is based upon work supported by
the National Science Foundation under agreement 0112050. The second author was supported by a
NATO postdoctoral fellowship and by KBN 2-P03A-002-23.
http://www.siam.org/journals/siap/65-1/43172.html
† Department of Mathematics, Iowa State University, Ames, IA 50011 (halevine@iastate.edu, joannar@iastate.edu).
‡ Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-956 Warsaw, Poland
(jr@impan.gov.pl).
336
SINGULARITY FORMATION IN CHEMOTAXIS
337
This system is not of standard reaction-diffusion type since the first equation
will involve the Laplacian of the second dependent variable. Typically, the boundary
conditions are of Neumann type for the first equation and of either mixed or Dirichlet
type for the second equation. Precise conditions will be given later.
In the growing literature on singularity formation in chemotaxis, the problems
studied tend to fall into one of two types.1
In the first type, the time scale of the second equation is assumed to be much
smaller than that of the first, i.e., D2 D1 , or the cell species has infinite propagation
speed, while the chemical species has diffused so rapidly that it has come to a steady
state. In this case, the system simplifies into one of elliptic-parabolic type, namely,
(1.2)
ut = D1 (∆u − ∇ · (uχ(v)∇v)),
0 = D2 ∆v + λu − av.
A number of papers have been concerned with the phenomenon of blowup for such
systems. See [2, 1, 5, 9, 10, 8, 19], for example. The first such result in this direction
seems to be contained in [13]. The rough idea of the approach to this system is to solve,
at least in principle, the second (elliptic) equation in (1.2) for v as a nonlocal, but
linear, function of u and then eliminate v from the first equation leaving a nonlinear,
nonlocal dynamical equation for u.
In the second, and probably less well studied form, the time scale of the second
equation is assumed to be much larger than that of the first, so fast in fact that
the diffusion of the chemical species can be neglected. (That is, D2 D1 .) In this
case the spatial movement of the chemical is being controlled by the movement of
the particles which the chemical influences through its gradient. This was used as an
example to illustrate the modelling approach to Dictyostelium discoideum movement
taken in [21].
The reaction term λu−av is the cAMP saturated limit approximation of a reaction
term that more accurately reflects Michealis–Menten reaction kinetics. A model of
D. discoideum movement which views the cell receptors as the catalyzing agent for
cAMP production can be found in [6, pp. 498ff.]. However, if cAMP is not in excess,
uv
then one must replace the reaction term by a term of the form kk21+v
− av, where
λ = k1 /k2 , in order to more accurately describe the cell receptor kinetics involved
(see [6, pp. 273ff.]). The system then takes the form
u
,
ut = D1 ∇ · u∇ ln
ψ(v)
(1.3)
k1 uv
− av.
vt = R(u, v) =
k2 + v
The advantage of writing the system in this form is that one can see that if the system
is tending toward a steady state, u should follow ψ(v).
Indeed, by using the principles of reinforced random walk [4], the authors of [21]
derived the first of equations (1.3) ab initio. However, by writing the flux vector
−
→
J as
−
→
−
→
−
→
J = −D1 ( ∇ u − χ(v)u ∇ v),
1 If one sets τ = D t, then D drops out of the first equation, while the second equation becomes
1
1
D1 ∂τ v = D2 ∆v + λu − av. One then lets D1 → 0 in this equation in the first case. In the second
case, one lets D1 , a, λ → ∞ in such a way that a/D1 , λ/D1 remain constant.
338
HOWARD A. LEVINE AND JOANNA RENCLAWOWICZ
defining ψ(v) by the equation ψ (v)/ψ(v) = χ(v), and using the continuity equa−
→ −
→
tion ut = − ∇ · J , we obtain the first equation in (1.3) by continuum mechanical
considerations.
Here the solution approach, taken, for example in [17], is to solve the second
equation for u as a nonlinear function of v, vt and then eliminate it from the first
equation, leaving a rather messy third order equation in v. Reference [17] is devoted
to a detailed discussion and interpretation of the numerical results obtained there and
earlier in [21] for the resulting equation.
In [17], the discussion begins with consideration of the following special case of
(1.3) on an interval [0, 1] for t > 0 with ψ(v) = v:
u
,
ut = ∇ · u∇ ln
v
(1.4)
vt = R(u, v) = uv
with ux (0, t) = ux (1, t) = 0 and vx (0, t) = vx (1, t) = 0 (which then imply the zero flux
conditions ux = uvx at x = 0, 1). The vector [µ(x, t), v0 (x, t)]t ≡ [1, et ]t is a spatially
homogeneous solution of (1.4) with [1, 1] as initial datum. The following statement is
a consequence of the results of [17]: Let c be the positive root of c2 +N c−1 = 0 and let
0 < < 1. Given any mode number N , there is a direction [uN , vN ]t ≡ [N c, 1]t cos(N x)
in the closed subspace of L2 (0, 1)×H 1 (0, 1) consisting of the closure of functions which
−
→
satisfy u[log(u/v)]x = 0 at x = 0, 1, and a curve given by R (ε) ≡ [u(·, 0, ε), v(·, 0, ε)]t
in L2 (0, 1) × H 1 (0, 1) of initial data passing through [1, 1] with the property that any
solution initially emanating from this curve will blow up in a finite time.
This solution is given by u = ψt (x, t), v = exp(ψ), where
ψ(x, t) = t − ln[1 − 2ε exp(N ct) cos(N πx) + ε2 exp(2N ct)].
Moreover, this solution has the important biological property that it leaves the above
space by aggregation, in particular, by virtue of the fact that u(·, t)L2 (0,1) blows
up in finite time. It is conceivable that for such systems, u(·, t)L2 (0,1) can remain
bounded, while ux (·, t)L2 (0,1) blows up in finite time. In [21], this possibility was
demonstrated numerically, while in [17], a plausibility argument was given to show
that these numerical results were not just artifacts of the simulations and were to be
expected from the underlying dynamical system.
The result tells us that in every neighborhood of the initial data for the spatially
homogeneous solution [1, et ]t , there are solutions of arbitrarily high initial total variation which begin in this neighborhood and blow up in finite time. The numerical
evidence suggests that every arbitrarily small nonconstant perturbation of the initial
data for [1, et ]t (which must have a nontrivial projection onto at least one of the directions [N c, 1]t cos(N x) for some N ) must blow up in finite time. (This interpretation
was not spelled out in [17].)
Clearly if we replace v by exp(v) in the system (1.4), there results
(1.5)
ut = uxx − (uvx )x ,
vt = u,
which, when ≥ 0, is a special case of
(1.6)
ut = uxx − (uvx )x ,
vt = vxx + u − av.
SINGULARITY FORMATION IN CHEMOTAXIS
339
Equation (1.6) is the classical system studied in [3]. In its turn, this system contains as a special case the system of Nagai and Nakaki taken up in section 2. However,
it is important to note that in [20], the authors establish the well posedness of the
initial-boundary value problem for this system (with homogeneous Neumann boundary conditions) as well as the existence of a global attractor. Their proof demands
that > 0 in order to establish the existence of a global Lyapunov functional. An
alternate proof of this result has been given in [12]. There the authors also provide
an asymptotic profile of the solution.
Thus we are left with the question of what happens to solutions when the dissipation in v is weak, i.e., when = 0 and a > 0. This is the problem raised by Nagai
and partially addressed by him and Nakaki in [18].
The plan of the paper is as follows. In section 2 we discuss their system and
the results they established recently [18]. There we also discuss a related initial
value problem for their system and introduce a closely related approximate initial
value problem. In section 3 we reformulate the Nagai–Nakaki system as an infinite
system of nonlinear ordinary differential equations. In section 4, we introduce a second
infinite system of ordinary differential equations which is closely related to the system
of ordinary differential equations in section 3. This second system is related to the
first in much the same way as the approximate initial value problem is related to the
full initial value problem for their system.
In section 5 we establish the local existence and uniqueness of solutions of their
system when µ > a in the sequence space 1 ×11 , i.e., in the space of pairs of sequences
({ai }, {bi }) such that i≥1 (|ai | + i|bi |) is finite. (This sequence space is continuously
and injectively imbedded in L1 (0, 1)×W 1,1 (0, 1). However, the inverse of the injection
(restricted to the image) is not continuous. We discuss this point in more detail in
section 7.)
In section 6, we demonstrate that the spatially homogeneous solutions of the
system of ordinary differential equations for the approximate problem are unstable in
the sense that in every neighborhood of the spatially homogeneous solution there are
solutions in L2 (0, 1) × H 1 (0, 1) with spatially inhomogeneous data which blow up in
finite time in this space.
In section 7, we establish this conjecture. That is, if µ > a, then in every neighborhood of (µ, v0 ) there are (spatially nonconstant) initial data for which the corresponding solution of the Nagai–Nakaki problem in the cylinder must blow up in finite
time (in a sense to be made precise below). This result will yield the Nagai–Nakaki
conjecture for solutions that are sufficiently regular.
2. The system of Nagai and Nakaki [18]. Nagai, in a talk given at the
International Conference on Partial Differential Equations and Mathematical Biology,
Wuhan, China, 2001, considered the following initial-boundary value problem:
(2.1)
ut = uxx − (uvx )x ,
vt = u − av
with a ≥ 0. As boundary conditions he took
(2.2)
ux (0, t) = vx (0, t)u(0, t),
ux (1, t) = vx (1, t)u(1, t).
These boundary conditions follow from the conditions ux (0, t) = ux (1, t) = 0 and
vx (0, 0) = vx (1, 0) = 0 because the second equation of (2.1) implies that vx (0, t), vx (1, t)
340
HOWARD A. LEVINE AND JOANNA RENCLAWOWICZ
satisfy y (t) = −ay(t). (In what follows we will refer to (2.1) and (2.2) as Nagai’s problem or the Nagai–Nakaki problem.)
To set the notational stage for what follows, we introduce a potential function
ψ(x, t) as follows: We let v(x, t) ≡ V (t)+ψ(x, t), u(x, t) ≡ U (t)+ ũ(x, t). The spatially
homogeneous solution of Nagai’s problem is
(V (t), U (t)) = (µ/a + (v0 − µ/a) exp(−at), µ).
This and the second equation force the choice for ũ = ψt + aψ so that
(2.3)
(ψ, ψt + aψ)t = (v − V (t), u − µ) = (v, u)t − (V (t), U (t))t .
In particular, in what follows, the reader is cautioned that (ψ, ψt + aψ)t corresponds
to the pair (v, u + aψ)t . (That is, with reference to [18], the pairing is (u, v), while in
the theorems and proofs here, the pairing is (v, u).)
We are interested in those potential functions for which
1
1
(2.4)
ψt dx = 0
ψ dx =
0
0
1
in order to ensure that the mass, 0 u dx, is conserved.
Recently Nagai and Nakaki [18] have established the following statements for a
restricted class of initial data perturbations for which the corresponding solutions are
more regular than L2 (0, 1) × H 1 (0, 1).
To save the reader a bit of time, we give a rough summary of their results. (By
initial values, we mean initial values for which the mean value of u is µ and which are
not identical with the stationary solution (µ, µ/a).)
1. If the initial values are sufficiently regular (in particular if they are analytic functions) and if they satisfy the boundary conditions, then the solution
components u, v will be in H 2 (0, 1) and will be continuous or continuously
differentiable in time into the appropriate range on the interval of existence.
2. If µ < a and the initial values are sufficiently regular, then (u, v) approaches
(µ, µ/a) exponentially rapidly in the norm of H 1 (0, 1) × H 2 (0, 1).
3. Suppose µ > a and suppose that the initial values are sufficiently regular.
a. If the initial data satisfy
W (u(, ·, 0), v(·, 0)) < µ ln µ −
µ2
,
2a
where f > 0, g ≥ 0 and where
1
W (f, g) ≡
(f ln f − f g + ag 2 /2) dx,
0
and if the solution exists for all time, then each component of (u, v)
blows up in the H 1 norm as t → +∞.
b. If the blow-up time is finite, then each component blows up in L∞ , i.e.,
pointwise.
3. There are also solutions in each such neighborhood which converge to the
steady state solutions in infinite time in the norm of H 1 (0, 1) × H 2 (0, 1).
Stability results for related problems have been established in [7, 23]. All of the
exact solutions found in [23] were found earlier in [17] in rescaled form, contrary to
341
SINGULARITY FORMATION IN CHEMOTAXIS
the implication in [23, p. 776]. (The solutions in [23] follow from those of [17] after a
shift in the time axis.)
In his talk, Nagai mentioned that he and Nakaki were unable to resolve whether
or not the blowup occurred in finite time. Therefore, we shall refer to the following
statement as Nagai’s conjecture. There are choices of initial data (u(·, 0), v(·, 0)) for
which the blow-up time must be finite.
In view of the results of [17], when a = 0, the blowup must occur in finite time.
Motivated by this simple observation, we set about trying to establish this for Nagai’s
problem. However, we were unable to establish this claim in L2 (0, 1) × H 1 (0, 1).
In the course of our investigations, we happened upon a problem, which is, in a
sense, close to that of the problem of Nagai, for which Nagai’s conjecture holds for
(u, v) ∈ L2 (0, 1) × H 1 (0, 1). Moreover, this result allows us to prove the conjecture of
Nagai in a Banach space different from that proposed by Nagai. More precisely, if we
(t), bn (t))}∞
denote by {(an
n=1 the sequence of cosine coefficients for the pair (ψ, ψt ),
we show that n (n|an (t)| + |bn (t)|) must blow up in finite time.
As remarked above, blowup in sequence space in this sense does not imply blowup
of the L1 norm of ψ, ψx , or ψt .
We see that ψ satisfies
ψtt + (µ − a)ψxx = (ψtx − ψt ψx )x − a(ψt + (ψψx )x ).
(2.5)
In order to motivate the approximate problem, we digress for a moment and
consider the pure initial value problem for (2.5). If we compute the Fourier transform
t) = ∞ e−iξx ψ(x, t)dx and assume that ψ, ψx vanish at x = ±∞
ϕ(ξ, t) = ψ(x,
−∞
on any interval [0, T ) where the solution of the initial value problem exists, we find
(suppressing the second argument on the right)
aξ 2
ϕ ∗ ϕ(ξ) + ξϕt ∗ (ηϕ)
2
ξ
1
[aξ 2 ϕ(ξ − η)ϕ(η) + 2ξ(ξ − η)ϕ(ξ − η)ϕt (η)] dη
=
2 0
1 ∞
+
{aξ 2 ϕ(ξ − η)ϕ(η) + ξ[(ξ − η)ϕ(ξ − η)ϕt (η)
2 ξ
ϕtt + (a + ξ 2 )ϕt + ξ 2 (a − µ)ϕ =
(2.6)
+ ηϕ(η)ϕt (ξ − η)]} dη.
If ψ, ψx are sufficiently regular, then
(2.7)
lim
|ξ|→+∞
∞
aξ 2 ϕ(ξ − η)ϕ(η) + ξ[(ξ − η)ϕ(ξ − η)ϕt (η) + ηϕ(η)ϕt (ξ − η)] dη = 0.
ξ
We can estimate the terms in (2.7) as follows:
∞
2
≤ ξ 2 ϕL∞
ξ
ϕ(ξ
−
η)ϕ(η)
dη
ξ
∞
|ϕ(η)| dη,
ξ
while
1 ∞
ξ[(ξ − η)ϕ(ξ − η)ϕt (η) + ηϕ(η)ϕt (ξ − η)] dη ≤ |ξ|ηϕ(η)L∞
2
ξ
ξ
∞
|ϕt (η)| dη.
342
HOWARD A. LEVINE AND JOANNA RENCLAWOWICZ
These inequalities give us an idea of how rapidly the transform of the solution should
decay.
The Fourier transform of the partial differential equation which approximates
Nagai’s problem is the equation
(2.8)
ϕtt + (a + ξ 2 )ϕt + ξ 2 (a − µ)ϕ
1 ξ 2
=
[aξ ϕ(ξ − η)ϕ(η) + 2ξ(ξ − η)ϕ(ξ − η)ϕt (η)] dη ≡ Φ(φ, φt )(ξ, t).
2 0
The nonlinear partial differential equation itself can be recovered from (2.8) in
the form
(2.9)
ψtt + [aψ + ψxx ]t + (µ − a)ψxx = Φ (φ, φt )(x, t),
where Φ denotes the inverse Fourier transform of Φ. We call Φ the finite part of
the Fourier transform of (uvx )x . (Notice that, except for the factor ξ 2 , Φ is the sum
of the finite parts of two convolutions, one of φ with itself and the other of φt (ξ) with
ξφ(ξ).)
In section 4, we derive a system of ordinary differential equations for the cosine
coefficients of the initial-boundary value problem for (2.9). This initial-boundary
value problem corresponds to the natural initial-boundary value problem for (2.5)
which arises from Nagai’s problem, i.e., ψx = ψxt = 0 at x = 0, 1 and t > 0 with
ψ, ψt prescribed at t = 0. The resulting system of ordinary differential equations for
the cosine coefficients of the solution of (2.9) is related to the corresponding system
of ordinary differential equations for the cosine coefficients of the solution of Nagai’s
problem in much the same way that (2.8) is related to (2.6).
In order to see how the former system comes about, we next derive the corresponding system of ordinary differential equations for Nagai’s problem.
3. Reformulation of the Nagai–Nakaki system as a system of ordinary
differential equations. We introduce some notation. Let β ≥ 0 and i ∈ {1, 2}.
We work in the spaces iβ ([0, T )) of sequences of real valued functions {gn (t)}∞
n=1 on
∞
[0, T ) for which n=1 nβ |gn (t)|i < ∞. When β = 0 we omit the subscript. That is,
iβ = i0 = i .
i
We say that a sequence of differentiable functions {gn (t)}∞
n=1 is in β ([0, T )) ×
i
∞
jβ ([0, T )) if the sequence {gn (t)}∞
n=1 is in β ([0, T )) and the sequence {gn (t)}n=1 is
j
in β ([0, T )).
Assuming that µ > a, we seek a solution of this equation in the form
(3.1)
ψ(x, t) =
∞
gn (t) cos(Cnx),
n=1
where C = 2πM for some integer M . With this choice of M the conservation conditions (2.4) hold.
SINGULARITY FORMATION IN CHEMOTAXIS
343
Consequently,
(ψt ψx )x = −C
∞ kgk gl (cos(Clx) sin(Ckx))x
n=2 k+l=n
(3.2)
∞ 1
kgk gl [(k + l) cos(C(k + l)x) + (k − l) cos(C(k − l)x)]
= − C2
2
n=2
k+l=n
∞ n−1
1
= − C2
kgk gn−k
[n cos(Cnx) + (2k − n) cos(C(2k − n)x)],
2
n=2
k=1
and
(ψψx )x = −C
(3.3)
∞ kgk gl (cos(Clx) sin(Ckx))x
n=2 k+l=n
∞ 2
1
=− C
2
kgk gl [(k + l) cos(C(k + l)x) + (k − l) cos(C(k − l)x)]
n=2 k+l=n
∞ n−1
1 2
=− C
gk gn−k [n2 cos(Cnx) + (2k − n)2 cos(C(2k − n)x)].
4
n=2
k=1
Then (2.5) can be rewritten in the form
∞
(gn + (C 2 n2 + a)gn − (µ − a)C 2 n2 gn ) cos(Cnx)
n=1
∞ n−1
1 2
C
{kgk gn−k
[n cos(Cnx) + (2k − n) cos(C(2k − n)x)]
2
n=2 k=1
a
+ gk gn−k n2 cos(Cnx) + (2k − n)2 cos(C(2k − n)x) }.
2
The terms involving cos(C(2k − n)x) on the right-hand side can be rewritten by
switching the order of summation and setting l = 2k − n to obtain
=
∞
∞
∞ n−1
(2k − n)kgk gn−k
cos(C(2k − n)x) =
(2k − n)kgk gn−k
cos(C(2k − n)x)
n=2 k=1
k=1 n=k+1
=
=
∞ k−1
lkgk gk−l
cos(Clx)
k=1 l=−∞
∞
∞ (−l)kgk gk+l
cos(Clx)
k=1 l=1
+
=−
∞ k−1
k=1 l=1
∞ ∞
lkgk gk−l
cos(Clx)
lkgk gk+l
cos(Clx)
l=1 k=1
+
∞
∞ l=1 k=l+1
lkgk gk−l
cos(Clx).
344
HOWARD A. LEVINE AND JOANNA RENCLAWOWICZ
Likewise,
∞ n−1
gk gn−k (2k − n) cos(C(2k − n)x) =
2
n=2 k=1
=
∞
∞
gk gn−k (2k − n)2 cos(C(2k − n)x)
k=1 n=k+1
∞
∞
gk gl+k l2 cos(Clx)
k=1 l=−(k−1)
∞
∞ 2
l
gk gl+k cos(Clx),
=2
l=1
k=1
the third line following from the second by breaking up the inner sum on the right
of the second line into an infinite sum over positive integers and a finite sum over
negative integers l = −1, . . . , −(k − 1). The latter inner sum is then rewritten as a
finite sum over positive indices and the order changed in the resultant double sum.
Therefore, we have
∞
(gn +(C 2 n2 + a)gn − (µ − a)C 2 n2 gn ) cos(Cnx)
n=1
(3.4)
=
∞ n−1
1 2
a
C
nkgk gn−k
+ n2 gk gn−k cos(Cnx)
2
2
n=2 k=1
∞
∞
∞
∞
1 2
+ C
n
kgk gk−n −
kgk gk+n + an
gk gn+k cos(Cnx).
2
n=1
k=n+1
k=1
k=1
We obtain the following (infinite) system of ordinary differential equations:
Ln gn ≡ gn + (C 2 n2 + a)gn − (µ − a)C 2 n2 gn
n−1
1 2
a
kgk gn−k
= C n
+ ngk gn−k
2
2
(3.5)
k=1
∞
+
(n + k)gn+k gk − kgk gk+n + angk gn+k
for n = 1, 2, . . . .
k=1
In order to rewrite (3.5) in a more compact form, we introduce the notation
|g| = {|gk |}∞
k=1 ,
Tn g = {gn+k }∞
k=1 (shift operator),
∞
g = {gk }k=1 (differentiation),
Mg = {kgk }∞
(multiplication by the transform variable),
k=1
∞
n−1
gk hn−k
(convolution),
g∗h=
k=1
(g, h) =
∞
n=1
gk hk (scalar product in 2 ).
k=1
Then one can solve the Nagai–Nakaki system in the aforementioned function space
if and only if one can solve the initial value problem for the following system in the
SINGULARITY FORMATION IN CHEMOTAXIS
345
corresponding sequence space:
Ln gn ≡ gn + (C 2 n2 + a)gn − (µ − a)C 2 n2 gn
(3.6)
a
1 2
= C n (Mg ∗ g )n + n (g ∗ g)n + [(Tn Mg, g ) − (Mg, Tn g )] + an(g, Tn g) .
2
2
4. A system of ordinary differential equations related to the Nagai–
Nakaki system. If we consider (3.6) without the last three terms on the right-hand
side, we obtain the following system of ordinary differential equations:
Ln gn ≡ gn + (C 2 n2 + a)gn − (µ − a)C 2 n2 gn
n−1
a
1 2
kgk gn−k + ngk gn−k
= C n
2
2
k=1
1 2
a
≡ C n (Mg ∗ g )n + n (g ∗ g)n .
2
2
(4.1)
That is, (4.1) is the discrete version of (2.8). It is the system satisfied by the cosine
coefficients of solutions of (2.9).
Comparing (4.1) and (2.8), we infer that the three terms in (3.6) that do not
appear in (4.1) can be viewed as “tail ends” of integrals. That is, they are analogous
to the three integrals that have been dropped in passing from (2.6) to (2.8) under the
assumption that ψ, ψx are sufficiently regular.
This suggests that (Tn Mg, g ), (Mg, Tn g ), na(g, Tn g) can be neglected in comparison with the remaining terms on the right-hand side of (3.6).
Such a statement needs rigorous proof.
5. Local existence and uniqueness of solutions of the initial value problem for the Nagai–Nakaki system. We prove the following result in 11 × 1 which
was proved in [18] in a product of smoother spaces.
Lemma 1. The solution {gn (·)}∞
n=1 of the system (3.5) exists locally in time and is
unique in 11 ×1 on the interval of local existence. Moreover, the solution is uniformly
bounded in the 11 × 1 norm on compact subsets of the existence interval.
Proof. First consider the question of uniqueness. Set wn = gn − hn , where gn and
hn are two solutions of the above system for n ≥ 2 for which g(0) = h(0), g (0) = h (0).
The difference wn satisfies the equation
Ln wn ≡ wn + (C 2 n2 + a)wn − (µ − a)C 2 n2 wn
n−1
1 2
k(wk gn−k
= C n
+ hk wn−k
) + ak(wk gn−k + hk wn−k )
2
k=1
∞
(5.1)
+
(n + k)(wn+k gk + hn+k wk ) − k(wk gn+k
+ hk wn+k
)
k=1
+ 2a
∞
k(wk gn+k + hk wn+k ) ,
k=1
which, in the above notation, becomes
Ln wn =
1 2
C n{(Mw ∗ g )n + (Mh ∗ w )n + a[(Mw ∗ g)n + (Mh ∗ w)n ]
2
+ (Tn Mw, g ) + (Tn Mh, w ) − (Mw, Tn g )
− (Mh, Tn w ) + 2a[(Mw, Tn g) + (Mh, Tn w)]}.
346
HOWARD A. LEVINE AND JOANNA RENCLAWOWICZ
We abbreviate this as
Ln wn = Fn (w, w ),
(5.2)
where we suppress the dependence of the right-hand side on g, h, g , h for the moment.
The characteristic equation for each of the linear second order operators Ln is
r2 + (C 2 n2 + a)r − (µ − a)C 2 n2 = 0.
The roots rn+ , rn− are real with rn+ > 0 > rn− , with rn+ → 2(µ − a) ≡ r+ as n → ∞,
and with rn+ ≤ r+ for all n while rn− → −∞. Since the initial values for w vanish, the
solution of (5.1) can be written as
−
+
wn = An (Fn , t)ern t + Bn (Fn , t)ern t ,
(5.3)
where
(5.4)
t
−
+
−ern s Fn (s)
1
An (Fn , t) =
e−rn s Fn (s)ds,
ds = 2
2
2
2
2
W
(C n + a) + 4(µ − a)C n 0
0
t
t r+ s
−
−1
e n Fn (s)
ds = Bn (Fn , t) =
e−rn s Fn (s)ds.
2
2
2
2
2
W
(C n + a) + 4(µ − a)C n 0
0
t
Because
−
+
max{|Bn (Fn , t)ern t |, |An (Fn , t)ern t |} ≤
c rn+ t
e
n2
t
e−rn s |Fn (s)|ds
+
0
for some computable constant c, we have
(5.5)
Mw(t)1 =
∞
n|wn (t)| ≤
n=1
r+ t
≤e
∞
c rn+ t t −rn+ s
e
e
|Fn (s)|ds
n
0
n=1
t
∞
c
|Fn (s)|ds.
n
0 n=1
We need to estimate the terms in the last integral. For each index n, there are 10
sums arising from the 10 terms in the definition of Fn /n. After use of the convolution
inequality, we have that
∞
n=1
|(Mw ∗ g )n | ≤
∞ n−1
k|wk ||gn−k
| = |Mw| ∗ |g |1 ≤ Mw(t)1 g (t)1 .
n=1 k=1
The next three sums are bounded above by a constant multiple of w (t)1 Mh(t)1 ,
Mw(t)1 Mg(t)1 , and Mw(t)1 Mh(t)1 , respectively.
The terms involving Tn are a bit trickier to estimate. We have, for the first of
them,
∞
n=1
∞ ∞
∞
∞
∞
|(Tn Mw, g )| ≤
(n + k)|wn+k ||gk | =
|gk |
l|wl | ≤
|gk |Tk Mw1 .
k=1 n=1
k=1
l=k+1
k=1
347
SINGULARITY FORMATION IN CHEMOTAXIS
Since Tk Mw1 is decreasing in k, we obtain
∞
|gk ||Tk Mw|1 ≤ Mw(t)1 g (t)1 .
k=1
In a similar fashion, the remaining five sums are found to be bounded above by a
constant multiple of w (t)1 Mh(t)1 , Mw(t)1 g (t)1 , w (t)1 Mh(t)1 ,
Mw(t)1 Mg(t)1 , and Mw(t)1 Mh(t)1 , respectively.
Thus, for some constant B
t
+
Mw(t)1 ≤ Ber t (Mw(s)1 + w (s)1 )(Mg(s)1 + g (s)1 + Mh(s)1
0
+h (s)1 )ds.
Assume that g(·), h(·) are in 11 ([0, T ]) and g , h are in 1 ([0, T ]); i.e., on every compact
subset K of [0, T ) there is a finite constant M (K) such that
∞
∞
max
(5.6)
(k|gk | + |gk |),
(k|hk | + |hk |) < M (K).
k=1
k=1
There results
Mw(t)1 ≤ Ber
(5.7)
+
t
t
(Mw(s)1 + w (s)1 )ds
0
for some new computable constant B = B(M (K)).
Next we estimate w (t)1 . Using the representation formula (5.3) we obtain
t
+
1
rn+
(5.8) wn (t) =
ern (t−s) Fn (s)ds
2
2
2
2
2
(C n + a) + 4(µ − a)C n
0
t
−
ern (t−s) Fn (s)ds .
− rn−
0
Consequently, for some positive computable constants c, d we have, noting that
rn− ≈ −dn2 ,
+ t
t
r
r + (t−s)
−dn2 (t−s) |Fn (s)|
ne
e
|Fn (s)|ds +
(5.9) |wn (t)| ≤ c
ds .
n2 0
n
0
The first
√ term on the right-hand side is treated exactly as above. We note that
for c1 = 1/ 2ed, all positive integers n, and all s ≤ t,
2
c1
.
ne−dn (t−s) ≤ √
t−s
Using this in (5.9) and summing over n we obtain
t
∞
∞
t
|Fn (s)|
c1 |Fn (s)|
+ r + (t−s)
√
r e
ds +
(5.10) w (t)1 ≤ c
ds .
n2
n
t − s n=1
0
0
n=1
Since we have already estimated the integrand sums
tain
(5.11)
r+ t
w (t)1 ≤ c1 e
0
t
∞
n=1
|Fn (s)|
n
above, we ob-
c
√
(Mw(s)1 + w (s)1 ) ds .
t−s
348
HOWARD A. LEVINE AND JOANNA RENCLAWOWICZ
Setting ϕ(t) = Mw(t)1 + w (t)1 , we have, with c = c(T ) = (cc1 + B)er
t
ϕ(t) ≤ c
(5.12)
0
+
T
,
ϕ(s)
√
ds,
t−s
a Volterra integral inequality of Gronwall type with a weakly singular kernel. Thus, after an application of the Hölder inequality, it is easily shown that ϕq , with
1/p + 1/q = 1 and 1 < p < 2, satisfies a standard Gronwall inequality. That is,
ϕ (t) ≤ c
q
q
Thus, for φ(t) =
t
0
pq
t
q
q
p−2
t
ϕq (s) ds.
0
q
q
q
2
) p s p − 2 , we obtain
ϕq (s) ds and c̃(s) = cq ( 2−p
d
dt
Then
2
2−p
−
φ(t)e
t
0
c̃(s) ds
≤ 0.
t
c̃(s) ds
0 ≤ φ(t) ≤ φ(0)e 0
.
Since φ(0) = 0, it follows that φ(t) ≡ 0.
Consequently, ϕ(t) ≡ 0, and hence g ≡ h on the existence interval.
Next, consider local existence. Since most of the estimates needed here have been
worked through above, we will be brief.
Abbreviate (3.6) as Lg = F (g, g ), g(0) = g0 , g (0) = g0 . Then the homogeneous
solution G(t) with inhomogeneous initial values solves
LG = 0,
G (0) = g0 ,
G(0) = g0 ,
while the function H = g −G satisfies the nonlinear problem with homogeneous initial
data:
LH = F (G + H, G + H ),
H(0) = 0, H (0) = 0.
Thus it suffices to show that the integral equation H = L−1 F (G + H, G + H ) has
a fixed point on some interval [0, Texist ). We see from the variation of parameters
formula (5.3) that the components satisfy
+
−
Hn = An (Fn , t)ern t + Bn (Fn , t)ern t .
1
1
∞
Define the sequence of iterates {H k }∞
k=1 as follows with H (t) = {Hn (t) = 0}n=1 and,
for k ≥ 1,
H k+1 = L−1 F (G + H k , G + H k ).
To construct a fixed point, we need to show that sequence ({H k }, {H k }) is convergent in 11 × 1 . We do this by showing that we can apply the contraction mapping
principle to the sequence. The argument usually goes in two steps.
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SINGULARITY FORMATION IN CHEMOTAXIS
1. There is a small time interval [0, T ] such that the sequence (of sequences)
1
1
({H k }, {H k })∞
k=1 is uniformly bounded in the norm on 1 × .
2. This sequence is contracting on this (or possibly smaller) time interval; i.e.,
there exists a constant 0 ≤ θ < 1 such that
sup M(H k+1 (s) − H k (s))1 + (H k+1 − H k ) (s)1
0≤s≤T
≤ θ sup M(H k − H k−1 )(s))1 + (H k − H k−1 ) (s)1 .
0≤s≤T
To establish step 1, we find, as in the uniqueness proof, an inequality of the form
MH k+1 (t)1 + (H k+1 ) (t)1
t
1
√
(M((G + H k )(s))1 + (G + H k ) (s)1 )2 ds.
≤c
t−s
0
Because the roots rn± are bounded above, we can assume that
sup [0, T ](MG(s)1 + (G (s)1 ) ≤ G0 (T )
for some constant G0 depending only on T . Consequently, we have the estimate
√ MH k+1 (t)1 + (H k+1 ) (t)1 ≤ c t G20 + [MH k (t)1 + (H k ) (t)1 ]2 .
√
If we set Zn = sup[0,T ] [MH n (t)1 + (H n ) (t)1 ], then Zn+1 ≤ c T (G20 + Zn2 ). Let
G1 > 0 be any constant such that Z1 ≤ G1 . Then Zn+1 ≤ G1 for all n = 1, 2, 3, . . . ,
provided that T is so small that
√
T ≤
G1
.
c(G20 + G21 )
Since both G0 , G1 depend on the initial values for the free solution, we have the
desired a priori bound on the iterates. For step 2, we examine the difference:
L(H k+1 − H k ) = F (G + H k , G + H k ) − F (G + H k−1 , G + H k−1 ) ≡ F(W k , W k ),
where W k = H k − H k−1 and F depends in fact on H k , H k , H k−1 , G, G . That is,
Ln Wnk+1 =
1 2
C n{(MW k ∗ (G + H k ) )n + (M(G + H k−1 ) ∗ W k )n
2
+ a[(MW k ∗ (G + H k ))n + (M(G + H k−1 ) ∗ W k )n ]
+ (Tn MW k , (G + H k ) ) + (Tn M(G + H k−1 ), W k )
− (MW k , Tn (G + H k ) ) − (M(G + H k−1 ), Tn W k )
+ 2a[(MW k , Tn (G + H k )) + (M(G + H k−1 ), Tn W k )]}.
Consequently, as in the previous proof, after obtaining a similar expression for W k
we can then use the estimates in the previous part of the proof to derive the estimate
of the form
t
1
√
MW k+1 (t)1 + (W k+1 ) (t)1 ≤ c
(MW k (s)1 + (W k ) (s)1 ) ds,
t−s
0
350
HOWARD A. LEVINE AND JOANNA RENCLAWOWICZ
where
c = c(M(G + H k−1 )1 , G + H k 1 , (G + H k ) 1 ).
By step 1, we may assume that c is uniformly bounded above by a constant c̃(T, G(0),
G (0)) for a sufficiently small time interval [0, T ]. We proceed as in the previous proof,
using the Hölder inequality to obtain the Gronwall inequality. From this, we easily
show that on a sufficiently small time interval one can apply the contraction mapping
principle to the sequence ({H k }, {H k }) in 11 × 1 . We omit the details.
6. Local existence and blowup of solutions of the approximate system.
We turn next to the local existence theorem and blow-up theorem for the initial value
problem for the approximate system (4.1). We choose the special initial sequence
gn (0) = an , gn (0) = nλan , where an and λ are to be chosen in such a manner that
{gn (t) = an enλt }∞
n=1 is a solution of the approximate system which must blow up in
finite time.
In what follows, we adopt the following notation. Let M be a positive integer
such that 4π2aM 2 ≡ a∗ ≤ 1. Suppose that ε, δ are such that
0 < 2ε ≤ a1 ≤
λ
δ,
λ+a
where λ is given by (6.3) below.
We establish the following theorems.
Theorem 1 (local existence). Suppose µ > a and the sequence {an }∞
n=2 solves
the recurrence relation given by (6.1) below. Then the sequence {gn (t) = an enλt }∞
n=1
solves (4.1) on an interval [0, Te ], where Te ≥ T∗ = − lnλδ . Moreover, (ψ(·, t), ψt (·, t))
is in H 1 (0, 1) × L2 (0, 1) on [0, T∗ ), where (ψ(·, t)) is given by (3.1). The function ψ
is analytic on (0, 1) × [0, T∗ ).
Theorem 2 (finite time blowup). The function (ψ(·, t), ψt (·, t)) of the previous
theorem must leave H 1 (0, 1) × L2 (0, 1) in finite time T∞ ≤ − lnλε .
Before proving these theorems, we show that for n ≥ 2
λ(4π 2 M 2 n − a)(n − 1)an = 2π 2 M 2
(6.1)
n−1
[λ(n − k) + a]kak an−k .
k=1
This recurrence formula defines the sequence an . However, unlike the situation in
[11, 17], we cannot find a simple expression for the coefficients an in terms of a1 and
n. Nonetheless, we can find upper and lower bounds for the series which sum to the
related solutions found in [17] for the case a = 0. These estimates provide the necessary
comparison functions for the existence and blowup of ψ in H 1 (0, 1) × L2 (0, 1).
For convenience of notation, we use ξ = x−1/2 in (3.1) instead of x. Then ∂x = ∂ξ
and single point blowup at ξ = 0 corresponds to blowup at x = 1/2. Therefore, if
gn = an enλt and C = 2πM , equation (4.1) reads
∞
(6.2)
an {4π 2 M 2 λn3 + λ2 n2 − (µ − a)4π 2 M 2 n2 + aλn}enλt cos(2πM nξ)
n=1
2
= 2π M
2
∞
n=2
n
n−1
k=1
(λ(n − k) + a)kak an−k
enλt cos(2πM nξ).
SINGULARITY FORMATION IN CHEMOTAXIS
351
Comparing coefficients for n ≥ 2,
n−1
[λ(n − k) + a]kak an−k .
[4π 2 M 2 λn2 + n λ2 − (µ − a)4π 2 M 2 + aλ]an = 2π 2 M 2
k=1
For n = 1,
a1 {λ2 + λ(4π 2 M 2 + a) − (µ − a)4π 2 M 2 } = 0.
Since µ > a, the roots are real. Let
(6.3)
λ=
1 2 2
( (4π M + a)2 + 16π 2 M 2 (µ − a) − 4π 2 M 2 − a)
2
denote the positive root.2 Then the relation for an , n ≥ 2, simplifies to (6.1) as
claimed. With the values of λ, a∗ above, we have that
(6.4)
2λ(n − a∗ )(n − 1)an =
n−1
(λ(n − k) + a)kak an−k .
k=1
We are now in a position to prove the theorems. We begin with Theorem 1.
Proof. From (6.1), since n ≥ k + 1,
2λ(n − a∗ )(n − 1)an ≤
n−1
(λ(n − k) + a)kak an−k
k=1
≤ (λ + a)
n−1
k(n − k) ak an−k .
k=1
If a1 ≤ b1 , by induction it follows that an ≤ bn , where
n−1
λ+a k(n − k)bk bn−k .
2(n − a∗ )(n − 1)bn =
λ
k=1
Because n ≥ 2, we have a∗ ≤ n/2 and
n(n − 1)bn ≤
n−1
λ+a k(n − k)bk bn−k .
λ
k=1
Comparing this sequence with bn , it again follows that if b1 = b1 , then bn ≤ bn , where
n(n − 1)bn =
n−1
λ+a k(n − k)bk bn−k .
λ
k=1
2 As in [17], the choice of the negative root yields a global solution which converges to the spatially
homogeneous solution as t → +∞. Notice also that if µ < a, both roots have negative real part and
the constructed solution must not only be global, it must converge to the spatially homogeneous
solution, an observation consistent with the results of Nagai and Nakaki [18]. When µ = a, λ = 0
and the constructed solution will be global but will not converge to the spatially homogenous solution.
352
HOWARD A. LEVINE AND JOANNA RENCLAWOWICZ
n−1 n
This recurrence relation can be solved explicitly with bn = n1 λ+a
(b1 ) . For
λ
λ
λ δn
b1 = λ+a δ, we have bn = λ+a n . The sequence {bn } defines a convergent series of
the form
ψ(x, t) =
∞
bn enλt cos(2πM nξ) =
n=1
=−
∞
λ 1 n nλt
δ e cos(2πM nξ)
λ + a n=1 n
λ
ln[1 − 2δeλt cos(2πM ξ) + δ 2 e2λt ]
λ+a
for t < − lnλδ . Consequently, the upper bound for ψ in L2 holds by the comparison
of the coefficients an and bn of series for ψ and ψ. A similar norm estimate holds for
ψt , ψ̄t . Thus ψ exists for all t < T∗ . That is, the existence interval [0, Te ] ⊃ [0, T∗ ) or
Te ≥ T ∗ .
We now turn to the proof of Theorem 2.
Proof. To obtain the lower bound, note that if a1 ≥ c1 , then an ≥ cn , where the
cn satisfy
2(n − a∗ )(n − 1)cn =
n−1
k(n − k)ck cn−k .
k=1
Thus
2n(n − 1)cn ≥
n−1
k(n − k)ck cn−k + 2a∗ (n − 1)cn ≥
k=1
n−1
k(n − k)ck cn−k .
k=1
Hence if c1 ≥ c1 > 0, then cn ≥ cn , where
2n(n − 1)cn =
n−1
k(n − k)ck cn−k .
k=1
n
1
However, cn = n2n−1
(c1 )n . Setting c1 = 2ε, it follows that an ≥ cn = 2 εn .
The function
ψ(x, t) = 2
∞
1 n nλt
ε e cos(2πM nξ)
n
n=1
= − ln[1 − 2εeλt cos(2πM ξ) + ε2 e2λt ]
exists as long as t < − lnλε because the series converges absolutely and uniformly. The
function ψ blows up pointwise at those points for which cos(2πM ξ) = 1. Although we
cannot compare these functions pointwise, we can compare them in H 1 . To see this,
note that from Parseval’s identity, it follows that |ψx |L2 (t) blows up in finite time.
n
Again, from Parseval and the inequalities an ≥ 2 εn , we have
|ψx |L2 (0,1) (t) ≤ |ψx |L2 (0,1) (t)
and
|ψt |L2 (0,1) (t) ≤ |ψt |L2 (0,1) (t).
SINGULARITY FORMATION IN CHEMOTAXIS
353
Thus, ψ “blows up” at T∞ ≤ − lnλε in the sense that the functions t → |ψx |L2 (t) and
t → |ψt |L2 (t) cannot be locally bounded on [0, ∞) and hence (u, v) cannot remain in
H 1 (0, 1) × L2 (0, 1) for all time.
We next require that
2ε ≤ a1 ≤
λ
δ
λ+a
so that
ε≤
(6.5)
λ
δ.
2(λ + a)
Then the blow-up time must satisfy − lnλδ ≤ T∞ ≤ − lnλε .
To prove the last claim of the theorem, note that since u = µ + ψt + aψ, it follows
yet again from Parseval that
|u(·, t) − µ|L2 (0,1) =
∞
1
|an |2 (1 + n2 )e2nλt ≥ 4
∞
ε2n (1 + n−2 )e2nλt .
1
Consequently u must leave L2 in finite time. Notice that this blow-up time is at least
as large as the time of escape from H 1 .
Finally, note that ψ(x, 0) and ψt (x, 0) are uniformly bounded above by [2λ/(λ +
a)] ln(1 − δ) and [2λ/(λ + a)](λδ(2 + λδ)/(1 − δ)), respectively. Hence for sufficiently
small δ the initial values for the perturbed solution are positive and uniformly close
to those for the spatially homogeneous solution.
n
Corollary 1. If an ≥ 2 εn , the function (ψ(·, t), ψt (·, t)) in the theorem also
blows up in finite time in the sense that the sequence {(gn (t), gn (t))}∞
n=1 with gn (0) =
an and gn (0) = nan λ must leave 11 × 1 in finite time.
Since |ngn (t)| + |gn (t)| ≥ 4εn enλt , the result follows.
Remark 1. We give an argument in the next section that shows that in every
neighborhood of the spatially homogeneous solution, there are solutions of Nagai’s
problem, which, if they agree initially with solutions of the approximate problem and
are sufficiently regular, cannot be global. The key to this argument is the demonstration that one may neglect the terms which we have identified as “tail ends.”
To explain why this might be reasonable, if we evaluate (Tn Mg, g ), (Mg, Tn g ),
na(g, Tn g) for gn (t) = β n enλt for β ∈ (0, 1) and λ > 0, one has gn = nλgn so
that the sum of the three neglected terms is bounded above by a constant multiple
of β n+2 eλ(n+2)t /(1 − β 2 e2λt ), while the sum of the convolution terms behaves like
nβ n enλt . Therefore on any compact subinterval of [0, − ln β/λ) the terms involving
Tn are small in comparison to the convolution terms for all sufficiently large n. Elementary calculations with series for which gn (t), gn (t) ≈ A/n(2+δ) show that the
terms involving the Tn are not necessarily small when compared with the convolution
terms.
A loose interpretation of this is the following: The partial differential equation
ψtt + (µ − a)ψxx = (ψtx − ψt ψx )x − a(ψt + (ψψx )x ) can be written in the form
ψtt + [aψ − ψxx ]t + a(ψψx )x + (ψt ψx )x + (µ − a)ψxx = 0,
which can be viewed as a quasi-linear second order partial differential equation with
a strong damping term (aψ − ψxx )t . Suppose that µ > a. Linearizing this equation
about ψ = 0 yields ψtt + (aψ − ψxx )t + (µ − a)ψxx = 0, an equation which is of
354
HOWARD A. LEVINE AND JOANNA RENCLAWOWICZ
elliptic type in the second derivative terms. Without the damping term, the solutions
are very regular, but the initial-boundary value problem is highly unstable. Even
with the damping term, solutions of the linear equation can blow up in infinite time.
The introduction of the nonlinear terms a(ψψx )x + (ψt ψx )x can lend a hyperbolic
character to the problem and force blowup in finite time by a focusing effect. See [17]
for a discussion of this when a = 0.
7. Nagai’s conjecture. From the local existence and uniqueness theorem we
know that the cosine series for ψ, ψt satisfies the condition that
(Mh(s)1 + h (s)1 )
(7.1)
is uniformly bounded on [0, τ ] for all τ in the existence interval of the solution.
We also know that in every neighborhood of the homogeneous initial data, there
is a solution of the approximate problem (4.1) with spatially nonconstant initial data
for which the solution blows up in 11 × 1 in finite time.
We establish the following theorem.
Theorem 3 (Nagai’s conjecture in 11 × 1 ). Suppose µ > a. Then the corresponding solution of the Nagai–Nakaki problem, for which the cosine coefficients
agree initially with the cosine coefficients of the aforementioned approximate problem,
cannot be global. That is, Nagai’s conjecture (in our sense) holds; i.e., such spatially
inhomogeneous solutions become unstable by blowing up in finite time in 11 × 1 .
Proof. Suppose that h(t) ≡ {hn (t)}∞
n=1 satisfies h(0) = an , h (0) = nλan and
the system of ordinary differential equations (3.6) on some interval, say, [0, Tmax ). Let
∗
g(t) ≡ {gn (t)}∞
n=1 satisfy g(0) = an , g (0) = nλan and satisfy (4.1) on [0, T ). Then
∗
Tmax ≤ T , and the solution of Nagai’s problem must blow up in finite time Tmax in
11 × 1 .
Suppose we could show that, on any time interval [0, τ ) where {hn (t)}∞
n=1 exists
in the sense of the local existence theorem,
(7.2)
sup M(h(t) − g(t))1 + sup h (t) − g (t)1 < C(h, τ ),
[0,τ ]
[0,τ ]
where C(h, τ ) does not depend on g. Then this inequality, together with Theorem
2 and the triangle inequality, would lead to a lower bound for the 11 × 1 norm of
∞
{hn (t)}∞
n=1 in terms of the corresponding norm for {gn (t)}n=1 and thus would permit
1
the establishment of Nagai’s conjecture in the space 1 × 1 .
To this end, suppose that τ < T ∗ < Tmax . We need to estimate wn (t) = hn (t) −
gn (t) in the same fashion that we did in the proof of local existence and uniqueness
where once again, wn (0) = wn (0) = 0. Define, for any sequence {zn (t)},
1 2
a
C n{(Mz ∗ z )n + n (z ∗ z)n },
2
2
1
Hn (z, z ) = C 2 n{[(Tn Mz, z ) − (Mz, Tn z )] + an(z, Tn z)}.
2
Gn (z, z ) =
(7.3)
SINGULARITY FORMATION IN CHEMOTAXIS
355
Then
Ln wn = Gn (h, h ) − Gn (g, g ) + Hn (h, h )
= Gn (h, h ) − Gn (w + h, (w + h) ) + Hn (h, h )
1
= C 2 n{−(Mw ∗ h )n − (Mh ∗ w )n − a[(Mw ∗ h)n + (Mh ∗ w)n ]}
(7.4)
2
+ Hn (h, h )
≡ Kn (w, w ; h, h ) + Hn (h, h )
≡ Fn (w, w ; h, h ).
This is the value of Ln that replaces the right-hand side of (5.2), (5.4)–(5.8). Notice
that the terms in the definition of Kn can be estimated as in the local existence and
uniqueness theorem. That is,
|Kn (w, w ; h, h )(s)| ≤ (Mw(s)1 + w (s)1 )(Mh(s)1 + h (s)1 ).
Likewise, we have
∞
|Hn (h, h )|(s)
n=1
n
≤ max{a, 1}
∞ ∞
[|hk (s)|j|hj (s)|+|hk (s)|j|hj (s)|+|hk (s)|j|hj (s)|].
k=1 j=1
For such functions, the conservation conditions (2.4) hold. Consequently,
∞
|Hn (h, h )|(s)
≤ max{a, 1}(h(s)1 + h (s)1 )Mh(s)1
n
n=1
by using estimates similar to those used for the estimates on the tail-end terms in the
proof of uniqueness.
Thus we obtain an inequality of the form
Mw(t)1 + w (t)1
∞
t
A(Mw(s)1 + w (s)1 ) + n=1 |Hn (h, h )|(s)/n
√
(7.5) ≤
ds
t−s
0
t
A(Mw(s)1 + w (s)1 ) + B(Mh(s)1 + h (s)1 )Mh(s)1
√
ds
≤
t−s
0
for some constant A depending on τ, Mh1 , h 1 and for some constant B depending perhaps on τ but not on w, w , h, h .
This inequality, (7.1), and an application of the Gronwall inequality will give us
the estimate (7.2). Combining this observation with its consequence (7.2) and the
triangle inequality gives the result.
Remark 2. The injection I : 1 → L1 (0, 1) given by
I({an }∞
n=1 )(x) =
∞
an cos (n − 1)πx
n=1
is certainly continuous. However, the inverse is not. To see this, let
P (x, ) =
1 − 2
1
2 1 + 2 − 2 cos(πx)
356
HOWARD A. LEVINE AND JOANNA RENCLAWOWICZ
denote the Poisson kernel for 0 ≤ < 1. The Poisson kernel is nonnegative, satisfies
1
P (x, ) dx = 1 = P (·, )L1 ,
0
and has Fourier cosine series
∞
∞
1 n
an cos (n − 1)πx = +
cos nπx
2 n=1
n=1
whose coefficient sequence satisfies
{an }∞
n=1 1 =
1+
.
1−
Therefore, as increases to unity, the 1 norm of the coefficient sequence increases
without bound, while the L1 norms of P (·, ) remain bounded. (Indeed, they converge
in measure to Dirac measure).
1
If µ ≥ 0, then we know from the first equation of the system that 0 u(x, t) dx =
1
µ(x) dx and hence must remain in L1 (0, 1) on the existence interval. The second
0
equation tells us that the second component must likewise remain bounded in L1 (0, 1).
The remark tells us that it is possible for the solution to blow up in sequence space
in finite time but remain bounded in the H 1 × L1 norm. If one knew that the solution
components blow up in finite time in 2β × 2β , for large enough β, β , then Parseval’s
identity would tell us that the solution would blow up in H β (0, 1) × H β (0, 1). Thus
we have the following corollary.
Corollary 2. Let δ, δ > 0. Suppose a solution of Nagai’s problem blows up
in finite time T in 11 × 1 . If the solution components belong to H 3/2+δ (0, 1) ×
H 1/2+δ (0, 1) and are bounded on compact subsets of the existence interval in the
norm of this product space, then the solution blows up in H 3/2+δ (0, 1) × H 1/2+δ (0, 1)
in finite time no larger than T .
Proof. This result follows from Schwarz’s inequality and Parseval’s identity.
Remark 3. The corollary states that certain very smooth solutions of Nagai’s
problem cannot be global. That is, they must lose regularity in finite time.
We can use the results of [18] to establish Nagai’s conjecture.
Corollary 3. Suppose µ > a. Then in every neighborhood of the stationary
solution, there are solutions which blow up in finite time in the sense that the H 2 ×H 1
norm blows up in finite time.
Proof. In [18, Proposition 4.1] the authors prove that if the initial data for u, v
are sufficiently smooth (in particular, if they are analytic) and satisfy the boundary
conditions, then both components are continuous from [0, Texist ) into H 2 (0, 1), while
the first (corresponding to v in the notation of [18]) is continuously differentiable from
[0, Texist ) into H 2 (0, 1).
The initial data for the approximate problem which give a solution of the approximate problem that blows up in finite time are in fact analytic (the Fourier coefficients
are bounded above by Cnn for small ) and consequently must satisfy the smoothness criteria of the initial data needed for [18, Proposition 4.1]. If we take the same
initial values for solution of Nagai’s problem, its components must belong to the same
spaces.
Thus, by the preceding corollary, the solution cannot be global. This, together
with the preceding corollary and δ = δ = 1/2, establishes Nagai’s conjecture for
certain sufficiently smooth data in every neighborhood of the stationary data.
SINGULARITY FORMATION IN CHEMOTAXIS
357
Remark 4. These results say nothing about the pointwise finite time blowup for
the solution components.
λ
δ, where λ is given in (6.3).
Corollary 4. Suppose µ > a and 0 < ε ≤ a1 ≤ λ+a
Then both components of the solution constructed in the preceding corollary blow up
in L∞ .
Proof. In [18, Theorem 7.1], the authors show that if the global existence time is
finite, then both components blow up in L∞ .
An illustrative computation is given in Figures 3 and 4.
Remark 5. If we replace λ by the negative root of the quadratic it satisfies, then
we can use the arguments of section 6 to obtain solutions of the approximate problem
which decay uniformly and exponentially rapidly to zero. It is then possible, by
appropriately modifying the arguments in Theorem 3, to show that the corresponding
solution of the Nagai–Nakaki problem, for which the cosine coefficients agree initially
with the cosine coefficients of the aforementioned approximate problem, must be global.
That is, Nagai’s conjecture (in our sense) holds; i.e., such spatially inhomogeneous
solutions (ψ, ψt ) exist for all time and must converge to the steady state (µ/a, µ) in
11 × 1 as t → ∞. We omit the details.
8. Illustrative computations. One might well ask whether or not solutions of
(6.1) are asymptotically of the form An for some || ∈ (0, 1) in the sense that there
is ∈ [0, 1) such that lim an /n = A for some constant A. We have shown that the
solutions of (6.1) are bounded above and below by terms of this form but we have not
yet established the asymptotics. However, the computations below provide a powerful
argument for these asymptotics.
The dependence of the blow-up time on M, µ − a for the function ψ in Theorem
1 can be investigated numerically as follows. It is worth noting that λ → 0+ as
M → +∞ for fixed µ − a or µ − a → 0+ for fixed M ≥ 1.
We begin by examining the growth of the terms of the sequence defined by (6.1).
If we set an (a) = bn (a)σ n /n with b1 = 1 and σ > 0, it is not too hard to see that when
a = 0, bn = 1 for all n ≥ 1. However, when 0 < a < µ, the terms bn grow remarkably
rapidly. As an illustrative example, with M = µ = 1 and a = 0.6, b1 = 1, ln b1000 ≈
656.9587. In fact, numerical evidence suggests that ln bn ≈ 0.657616(n−1)(1+o(1/n)).
Let τ be this coefficient of n−1 (assuming it exists). If we take σ = exp(−τ (1+δ)) for
any small, positive δ, then the solution should blow up in finite time t = τ δ/λ. This
will be the case if one can prove that the asymptotics for the bn are as indicated by the
numerical evidence. The numerical evidence indicates that as a increases to µ from
below, τ increases without bound, and hence the blow-up time will increase without
bound also. This is to be expected. (See Figure 1.) Likewise, the numerical evidence
indicates that as the integer M is increased for fixed a, τ approaches a limiting value
(which is to be expected since as M increases, λ → 0). This corresponds to a∗ = 0 so
that the sequence behaves like the exact solution when a = 0. The solution for a = 0
has the smallest blow-up time possible for fixed ε, δ, µ and all a ≥ 0. (See Figure 2.)
In Figures 3 and 4 we present a numerical simulation for the Nagai–Nakaki problem with ut = D(uxx − (uvx )x ), vt = u − av for 0 < x < 1, t > 0 and zero flux
boundary conditions. We took D = 0.02, µ = 4.0, and a = 1.0. For initial values we
used u(x, 0) = µ − cos(2πx) where = 0.4 and v(x, 0) = v0 = µ/a. These figures
provide some evidence that the solution does blow up pointwise in finite time as well
as in the 11 × 1 norm.
358
HOWARD A. LEVINE AND JOANNA RENCLAWOWICZ
(The lines corresponding to a = 0.8, 0.9 do not continue due to exponential overflow.)
Fig. 1. Growth of coefficients given by (6.1) for variable a.
(The lines corresponding to M = 0.09, 0.1, 0.15, 0.2 do not continue due to exponential overflow.)
Fig. 2. Growth of coefficients given by (6.1) for variable M.
SINGULARITY FORMATION IN CHEMOTAXIS
Fig. 3. Partial density, a = 1.0, D = 0.02, ε = 0.25, µ = 4.0.
Fig. 4. Chemical density, a = 1.0, D = 0.02, ε = 0.25, µ = 4.0.
359
360
HOWARD A. LEVINE AND JOANNA RENCLAWOWICZ
Acknowledgments. The authors take pleasure in thanking Thomas Hillen,
Peter Palacik, and Hans Weinberger and the referees for their constructive comments which considerably improved the earlier versions of this manuscript. The authors also take pleasure in thanking Toshitaka Nagai for sending us [18], for bringing
to our attention Proposition 4.1 of that paper, and for clarifying a point that led to
Corollary 4.
REFERENCES
[1] P. Biler, Local and global solutions of a nonlinear nonlocal parabolic problem, in Nonlinear
Analysis and Applications (Warsaw, 1994), GAKUTO Internat. Ser. Math. Sci. Appl. 7,
Gakkōtosho, Tokyo, 1996, pp. 49–66.
[2] P. Biler, Global solutions to some parabolic-elliptic systems of chemotaxis, Adv. Math. Sci.
Appl., 9 (1999), pp. 347–359.
[3] S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis, Math. Biosci., 56 (1981),
pp. 217–237.
[4] B. Davis, Reinforced random walk, Probab. Theory Related Fields, 84 (1990), pp. 203–229.
[5] J. I. Diaz, T. Nagai, and J.-M. Rakotoson, Symmetrization techniques on unbounded domains: Application to a chemotaxis system on RN , J. Differential Equations, 145 (1998),
pp. 156–183.
[6] L. Edelstein-Keshet, Mathematical Models in Biology, The Random House/Birkhäuser Mathematics Series, Random House Inc., New York, 1988.
[7] M. A. Fontelos, A. Friedman, and B. Hu, Mathematical analysis of a model for the initiation
of angiogenesis, SIAM J. Math. Anal., 33 (2002), pp. 1330–1355.
[8] M. A. Herrero, E. Medina, and J. J. L. Velázquez, Self-similar blow-up for a reactiondiffusion system, J. Comput. Appl. Math., 97 (1998), pp. 99–119.
[9] M. A. Herrero and J. J. L. Velázquez, Chemotactic collapse for the Keller-Segel model, J.
Math. Biol., 35 (1996), pp. 177–194.
[10] M. A. Herrero and J. J. L. Velázquez, Singularity patterns in a chemotaxis model, Math.
Ann., 306 (1996), pp. 583–623.
[11] T. Hillen and H. A. Levine, Blow-up and pattern formation in hyperbolic models for chemotaxis in 1-D, Z. Angew. Math. Phys., 54 (2003), pp. 839–868.
[12] T. Hillen and A. Potapov, The one-dimensional chemotaxis model: Global existence and
asymptotic profile, MMAS, to appear.
[13] W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential
equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), pp. 819–824.
[14] E. F. Keller, Assessing the Keller-Segel model: How has it fared?, in Biological Growth and
Spread (Heidelberg, 1979), Lecture Notes in Biomath. 38, Springer, Berlin, 1980, pp. 379–
387.
[15] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,
J. Theoret. Biol., 26 (1970), pp. 399–415.
[16] E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theoret. Biol., 30 (1971), pp. 225–234.
[17] H. A. Levine and B. D. Sleeman, A system of reaction diffusion equations arising in the
theory of reinforced random walks, SIAM J. Appl. Math., 57 (1997), pp. 683–730.
[18] T. Nagai and T. Nakaki, Stability of constant steady states and existence of unbounded
solutions in time to a reaction-diffusion equation modelling chemotaxis, Nonlinear Anal.,
58 (2004), pp. 657–681.
[19] T. Nagai and T. Senba, Global existence and blow-up of radial solutions to a parabolic-elliptic
system of chemotaxis, Adv. Math. Sci. Appl., 8 (1998), pp. 145–156.
[20] K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), pp. 441–469.
[21] H. G. Othmer and A. Stevens, Aggregation, blowup, and collapse: The ABC’s of taxis in
reinforced random walks, SIAM J. Appl. Math., 57 (1997), pp. 1044–1081.
[22] C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953),
pp. 311–338.
[23] Y. Yang, H. Chen, and W. Liu, On existence of global solutions and blow-up to a system
of reaction-diffusion equations modelling chemotaxis, SIAM J. Math. Anal., 33 (2001),
pp. 763–785.